Unlike the term 'German cut', which means that the stones were cut in Germany, French cut stones aren't referring to the fact that they were cut in France. It refers to their shape and design. They can be recognized by the typical cross the crown facets depict. French cut stones are square or rectangular multifaceted stones. They derived from making optimal use of dodecahedral diamond crystals.
To cut the crown of a french cut diamond one of the tops of the crystal is ground down to create a table that sits diagonal to the sides of the crystal. At this point the remaining crystal faces form natural facets that only need slight modelling to make it a symmetrical cut as can be seen in the image on the right. The outline is squared and the pavilion is cut to 4 plain facets adjusting the angle of the original faces to allow a high light return. Varieties where the facets described above are divided in half to create more facets are common.
French cut diamonds date back to the beginning of the 1400's but they came into fashion in the 17th century where they were favored favored by royalty and nobility until the brilliant cut was first introduced. French cut diamonds regained popularity during the Art Deco Era where they complimented the regular, geometric designs brilliantly.
Its name is probably derived from the fact that it was more popular in France then anywhere else. You may encounter the term flat-bottomed French cut. This refers to stones without a pavilion (like a rose cut). Their crowns follow the style of the traditional French cut.